Quadrilateralssrushingoe.weebly.com/uploads/1/2/5/6/12564201/... · 6-1 Angles of Polygons I CAN… INTERIOR ANGLES OF QUADRILATERALS Classify polygons abased on their sides and angles

QUADRILATERALS Chapter 6

GEOMETRY Name _______________________________

Hour __________

2

E F

G H

55º

xº

xº

6-1 Angles of Polygons

I CAN… INTERIOR ANGLES OF QUADRILATERALS

Classify polygons abased on their sides and angles find and use the measures of interior and exterior angles of polygons

A _____________________ of a polygon is a segment that joins two

__________________ vertices.

Like triangles, quadrilaterals have both _____________ and

_______________ angles.

Interior Angles of a Quadrilateral: The sum of the measures of the

interior angles of a quadrilateral is 360º.

𝑚 1 + 𝑚 2 + 𝑚 3 + 𝑚 4 = 360°

Example 6-1-1: Interior Angles of a Quadrilateral

Find mF, mG, and mH.

RECALL

diagonal

vertexside

A

E

D

B

C

1

2

3

4

Number of Sides

Name of Polygon

3 4 5 6 7 8 9

10 12 n

Triangle Quadrilateral

Pentagon Hexagon Heptagon Octagon Nonagon Decagon

Dodecagon n-gon

3

RECALL POLYGONS

Polygon: A plane _____________ that meets the following conditions.

1. It is formed by ________ or more segments called ____________,

such that no _____ sides with a _____________ endpoint are

noncollinear.

2. Each side intersects exactly ___________ other sides, one at each

______________.

A polygon is ___________ if no line that contains a side of the polygon

contains a point in the interior of the polygon.

A polygon that is not convex is called __________________ or

_______________.

A polygon is ___________________ if all of its sides are congruent.

A polygon is ___________________ if all of its interior angles are

congruent.

A polygon is ______________ if it is _________________ and

___________________.

Example 6-1-2: Identifying Regular Polygons

Decide whether the polygon is regular.

a. b. c.

_____________ ___________________ ________________

Polygon

Interior Angle

Sum Theorem

The sum of the interior angle measures of an

n–sided convex polygon is (n – 2) ⋅ 180.

Example 6-1-3: Find the sum of the measures of the interior angles

of each convex polygon.

A. octagon n = ___ B. dodecagon n = ___

4

(𝑛 – 2)180

𝑛

Find the measure of the exterior angle by subtracting the interior angle from 180. Exterior total is 360° Divide 360 by the exterior angle measure to know how many angles/sides in the polygon.

Example 6-1-4: Find the measure of each interior angle of pentagon

ABCDE.

Example 6-1-5: Find the measures of each interior angle of each

regular polygon.

A. Decagon n = _____ B. Heptagon n = ____

Example 6-1-6: The measure of an interior angle of a regular

polygon is 135. Find the number of sides in the polygon.

Polygon

Exterior Angle

Sum Theorem

The sum of the exterior angle measures of a convex

polygon, one angle at each vertex, is 360.

Example 6-1-7: Finding Exterior Angle Measures in Polygons

A. Find the value of 𝑏 in polygon FGHJKL.

B. Find the measure of each exterior angle of a regular dodecagon.

m∠H =_____° m∠J =_____° m∠K =_____°

m∠HL =_____° m∠F =_____° m∠G=_____°

5

6-2 Parallelograms

I CAN… PARALLELOGRAMS

prove and apply properties of parallelograms

use properties of parallelograms to solve problems

Any polygon with four sides is a

______________________________. However,

some quadrilaterals have special

properties.

These special quadrilaterals are given their own names.

Parallelogram →

Use the symbol for parallelogram.

Properties of Parallelograms

If a quadrilateral is a parallelogram, then …

_____________________________ are congruent.


_____________________________ are congruent.


_____________________________ are

supplementary. x + y = 180

If a quadrilateral has one right angle, then …

it has ____ right angles.


the __________________ bisect each other.


a ___________ _____________ cuts the

parallelogram into 2 triangles.

CB

A D

6

Example 6-2-1: In parallelogram CDEF, 𝐷𝐸 = 74 𝑚𝑚, 𝐷𝐺 = 31 𝑚𝑚, and

𝑚∠𝐹𝐶𝐷 = 42˚. Find each measure.

A. 𝐶𝐹 = _____

B. 𝑚∠𝐸𝐹𝐶 = _____

C. 𝐷𝐹 = _____

Example 6-2-2: Using the Properties of Parallelograms

GHJK is a parallelogram. Find the unknown length.

a. JH = ____ b. LH = ____

Example 6-2-3: Using the Properties of Parallelograms

In parallelogram ABCD, mC = 105º. Find the angle measure.

a. m A = _____ b. m D= _____

Example 6-2-4: Using Algebra with Parallelograms

WXYZ is a parallelogram. Find the value of x.

G H

J K

6 8

Y

Z W

3x + 18

4x - 9 X

L

A

B C

D

7

Example 6-2-5: 𝑾𝑿𝒀𝒁 is a parallelogram. Find each measure.

A. 𝑌𝑍 = _____ B. 𝑚∠𝑍 = ______

Example 6-2-6: 𝑬𝑭𝑮𝑯 is a parallelogram.

Find each measure.

A. 𝐽𝐺 = ______ B. 𝐹𝐻= ____

Example 6-2-7: Parallelograms in the Coordinate Plane

A. Three vertices of parallelogram

JKLM are 𝐽(3, −5), 𝐾(−2, 2), and

𝐿(2, 6). Find the coordinates of

vertex M.

B. Three vertices of parallelogram

PQRS are 𝑃(−3, −1), 𝑄(−1, 4), and

𝑆(5, 0). Find the coordinates of

vertex R.

M ( , ) R ( , )

8

6-3 Tests for Parallelograms I can… Prove that a given quadrilateral is a parallelogram.

You have learned to identify the properties of a parallelogram. NOW you

will be given the properties of a quadrilateral and will have to tell if the

quadrilateral is a parallelogram.

You can do this by using the definition of a parallelogram OR the

conditions below.

Parallelogram → both pair of __________________________ are parallel.

Conditions For Parallelograms

theorem example

If opposite sides are __________________

and congruent,

then the quadrilateral is a parallelogram.

If _________ ____________ of opposite

sides are congruent,


If opposite __________ are 𝒄𝒐𝒏𝒈𝒓𝒖𝒆𝒏𝒕

in a quadrilateral,


If a quadrilateral’s __________________

bisect each other,


Example 6-3-1: Verifying Figures are Parallelograms

A. Show that 𝐽𝐾𝐿𝑀 is a

parallelogram for 𝑎 = 3 and 𝑏 = 9

B. Show that 𝑃𝑄𝑅𝑆 is a

parallelogram for 𝑎 = 2.4 and 𝑏 = 9

1

2

3

4

9

A B

C D

1 2

3 4

REASONING ABOUT PARALLELOGRAMS

Example 6-3-2: Proving Facts about Parallelogram

Given: ABCD is a Parallelogram

Prove: 2 4

Statements Reasons

1.

1. Given

2.

2. Def of Parallelogram

3. 3. Corresponding

4. 4. Alt. Interior Angles

5. 2 4 5.

Example 6-3-3: Applying Conditions for Parallelograms

Determine if each quadrilateral must be a parallelogram. Justify your

answer.

A. B. C.

D. E. F.

G. H. I.

4

4 3

3

5

5

10

Example 6-3-4: Proving Parallelograms in the Coordinate Plane –

Show that quadrilateral 𝐽𝐾𝐿𝑀 is a parallelogram by using the given

definition or theorem.(opposite sides parallel)

A. 𝐽(−1, −6), 𝐾(−4, −1), 𝐿(4, 5), 𝑀(7, 0); definition of parallelogram

? ?

𝐾𝐽̅̅ ̅ ‖ 𝑀𝐿̅̅ ̅̅ 𝐾𝐿̅̅ ̅̅ ‖ 𝐽𝑀̅̅ ̅̅ ?

B. 𝐴(2, 3), 𝐵(6, 2), 𝐶(5, 0), 𝐷(1, 1): quad. with

pair of opp. sides ∥ and ≅ → parallelogram

? ?

AD = BC 𝐴𝐷̅̅ ̅̅ ‖ 𝐵𝐶̅̅ ̅̅

Example 6-3-2: Proving Facts about Parallelograms

Given: ACDF is a parallelogram. ABDE is a

parallelogram.

Prove: BCD EFA

Statements Reasons

1) 1) Given

2) 2) Opposite Sides ≅

3) 3)

4) 4) ≅ angles definition

5) 𝑚∠𝐹𝐴𝐵 = 𝑚∠𝐹𝐴𝐵 + 𝑚∠𝐹𝐴𝐵

𝑚∠𝐶𝐷𝐸 = 𝑚∠𝐶𝐷𝐵 + 𝑚∠𝐵𝐷𝐸 5)

6) 6) Substitution

7) 𝑚∠𝐹𝐴𝐵 − 𝑚∠𝐸𝐴𝐵 = 𝑚∠𝐶𝐵𝐷

𝑚∠𝐹𝐴𝐵 − 𝑚∠𝐸𝐴𝐵 = 𝑚∠𝐹𝐴𝐸 7) Subtraction

8) 8)

9) 9) ≅ angles definition

10) BCD EFA 10)

11

Section Summary – In each box sketch a parallelogram and label it to show how it meets

the conditions for a parallelogram

Conditio

ns fo

r

Parallelo

gram

s

12

6-4 Rectangles I can …

Prove and apply properties of rectangles. determine whether parallelograms are rectangles

WARM-UP: 𝑨𝑩𝑪𝑫 is a parallelogram. Find each measure.

1. 𝐶𝐷 = ______ 2. 𝑚∠𝐶 = _____

The first type of special quadrilateral we learned about is a

___________________________________.

A second type of special quadrilateral is a

______________________________.

Properties of Rectangles

Theorem Hypothesis

If a parallelogram is a rectangle, then…

If a parallelogram is a rectangle, then…

➢ Since a rectangle is a special type of a parallelogram, it “inherits”

all the properties of parallelograms that you learned in Lesson 6.2.

Example 6-4-1: Algebra with Rectangles

Quadrilateral RSTU is a rectangle. If 𝑚∠𝑅𝑇𝑈 = 8𝑥 + 4 and 𝑚∠𝑆𝑈𝑅 = 3𝑥 – 2, find x.

13

Example 6-4-2: Using Properties of Rectangles to Find Measures.

A. 𝐽𝐾𝐿𝑀 is a rectangular picture

frame. 𝐽𝐾 = 50 𝑐𝑚 and 𝐽𝐿 =86 𝑐𝑚.

Find 𝐻𝑀 = ______ and 𝐿𝑀 = _____

B. Quadrilateral EFGH is a

rectangle. If GH = 6 ft and FH =

15 ft.

Find GJ = ______ and FE = _____

Conditions For Rectangles

Theorem Example

If

then the parallelogram is a rectangle.

If

then the parallelogram is a rectangle.

Example 6-4-3: Rectangles in the Coordinate Plane

Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3).

Determine whether JKLM is a rectangle using the diagonals. Also verify

that the angles are right angles.

Step 1: Verify that it is a parallelogram

Step 2: Verify the lengths of the diagonals are congruent

Step 3: Verify the sides are perpendicular

14

6-5 Rhombi & Squares

I can…. Prove and apply properties of rhombi and squares. Determine whether parallelograms are rectangles, rhombi, or squares.

RHOMBI AND SQUARES

Rhombus Square

A ___ with _____ congruent

sides

A ___ with _____ congruent

sides and ____ right angles.

Rhombus Corollary: A __________ is a rhombus iff it has four ___________

__________.

Square Corollary: A __________ is a square iff it is a _________ and a

____________.

Properties of rhombi

THEOREM

If a parallelogram is a rhombus, then…

If a parallelogram is a rhombus, then…

➢ Like a rectangle, a rhombus is a parallelogram. So you can apply

the properties of parallelograms to rhombi.

➢ Therefore, a square has the properties of all three.

Example 6-5-1: Describing a Special Parallelogram

Decide whether the statement is always, sometimes, or never true.

a. A rectangle is a square. b. A square is a rhombus.

15

Example 6-5-2: Using properties of squares.

QRST is a square. What else do you know about QRST?

-Diagonals are ____ -4 sides are ____ -Diagonals bisect ___________

-Opposite sides _____ -4 right _______ -Diagonals are _____

Example 6-5-3: Using properties of rhombi.

A. 𝐴𝐵𝐶𝐷 is a rhombus. If 𝑚∠𝐴𝐵𝐶 = 126°, find 𝑚∠𝐶𝐷𝐵 . Also find 𝑚∠𝐶𝐹𝐷.

𝑚∠𝐶𝐷𝐵 = ____________ 𝑚∠𝐶𝐹𝐷 = ____________

B. 𝑇𝑉𝑊𝑋 is a rhombus. Find 𝑇𝑉 and 𝑚∠𝑉𝑇𝑍.

𝑇𝑉 = ____________ 𝑚∠𝑉𝑇𝑍 = ____________

C. 𝐶𝐷𝐹𝐺 is a rhombus. Find 𝐶𝐷 and 𝑚∠𝐺𝐶𝐻

if 𝑚∠𝐺𝐶𝐷 = (𝑏 + 3)˚ and 𝑚∠𝐶𝐷𝐹 = (6𝑏 − 40)˚.

𝐶𝐷 = ____________ 𝑚∠𝐺𝐶𝐻 = ____________

16

Example 6-5-4: Verifying Properties of Squares.

The vertices of square 𝑆𝑇𝑉𝑊 are 𝑆(−5, −4), 𝑇(0, 2), 𝑉(6, −3), and 𝑊(1, −9). Show that the diagonals of square 𝑆𝑇𝑉𝑊 are congruent perpendicular

bisectors of each other.

Step 1: Show that 𝑆𝑉 and 𝑇𝑊 are

congruent.

Step 2: Show that 𝑆𝑉 and 𝑇𝑊 are

perpendicular.

Step 3: Show that 𝑆𝑉 and 𝑇𝑊 bisect each other. (Use midpoint formula.)

Conditions for Rhombi

Theorem Example

Then a parallelogram is a Rhombus



17

Example 6-5-5: Applying Conditions for Special Parallelograms

Determine if the conclusion is valid. If not, tell what additional information

is needed to make it valid.

A. Given: 𝐸𝐹 ≅ 𝐹𝐺, 𝐸𝐺 ⊥ 𝐹𝐻

Conclusion: EFGH is a rhombus.

B. Given: ∠𝐴𝐵𝐶 is a right angle.

Conclusion: ABCD is a rectangle.

Example 6-5-6: Use the diagonals to determine whether a

parallelogram with the given vertices is a rectangle, rhombus, or

square. Give all the names that apply.

(Steps: Graph parallelogram; determine if rectangle, then rhombus, then

square.)

A. 𝑃(−1, 4), 𝑄(2, 6), 𝑅(4, 3), 𝑆(1, 1)

B. 𝑊(0, 1), 𝑋(4, 2), 𝑌(3, −2), 𝑍(−1, −3)

18

6-6 Trapezoids

I can… PROPERTIES OF TRAPEZOIDS

A ______________ is a quadrilateral with ___________ ________ pair of

__________ sides.

The __________ sides of a trapezoid are the ____________ of the trapezoid.

For each of the bases of a trapezoid, there is a pair of ______ ________,

which are the two __________ that have that base as a side.

The ____-________ sides of a trapezoid

are called __________.

If the _______ of a trapezoid are

congruent, then the trapezoid is an

__________ _________.

The ___________ of a trapezoid is the

segment that connects the midpoints of

its legs.

Isosceles Trapezoids

Theorem Hypothesis

If a trapezoid is isosceles, then…

If a trapezoid has

, then…

A trapezoid is isosceles iff it is

base angles

base angles

base

base

legleg

B

AD

C

19

Example 6-6-4: Using Properties of Isosceles Trapezoids

CDEF is an isosceles trapezoid with CE = 10 and mE = 95. Find DF,

mC, mD, and mF.

Example 6-6-5: Using Properties of Isosceles Trapezoids.

A. Find 𝑚∠𝐴 = _____ B. 𝐾𝐵 = 21.9 and 𝑀𝐹 = 32.7.

Find FB =____

Example 6-6-6: Applying Conditions for Isosceles Trapezoids.

A. Find the value of 𝑎 so that 𝑃𝑄𝑅𝑆 is isosceles.

B. 𝐴𝐷 = 12𝑥 − 11 and 𝐵𝐶 = 9𝑥 − 2. Find the value of 𝑥 so that 𝐴𝐵𝐶𝐷 is

isosceles.

F

C D

E

20

C. Find the value of 𝑥 so that 𝑃𝑄𝑆𝑇 is isosceles.

Trapezoid Midsegment Theorem

The midsegment of a trapezoid is parallel to each base and

its length is one half of the sum of

the lengths of the bases.

Example 6-6-8: Finding Midsegment Lengths of Trapezoids

A potter crafts a trapezoidal relish dish,

placing a divider, shown by AB , in the

middle of the dish. How long must the

divider be to ensure that it divides the

legs in half?

Example 6-6-9: Finding Lengths Using Midsegments

A. Find GH. B. Find EF. C. Find EH.

A B

8 in.

10 in.

10 A B

C D

G H

16

21

SPECIAL QUADRILATERALS - REVIEW I can…

Summarizing Properties of Quadrilaterals

quadrilateral

kite parallelogram trapezoid

rhombus rectangle

square

Example 1: Identifying Quadrilaterals

ABCD has at least two congruent consecutive sides. What quadrilaterals

meet this condition?

Example 2: Connecting Midpoints of Sides

When you join the midpoints of the sides of an isosceles trapezoid in

order, what special quadrilateral is formed why?

22

Three ways to prove a quadrilateral is a rhombus

1. You can use the definition and show that the quadrilateral is a

parallelogram that has four congruent sides. It is easier, however, to

use the Rhombus corollary and simplify show that all four sides of the

quadrilateral are congruent.

2. Show that the quadrilateral is a parallelogram and that the diagonals

are perpendicular.

3. Show that the quadrilateral is a parallelogram and that each diagonal

bisects a pair of opposite angles.

Example 3: Proving a Quadrilateral is a Rhombus

The coordinates of ABCD are A(-2, 5), B (1, 8), C (4, 5), and D (1, 2). Show

that ABCD is a Rhombus.

Parallelogram Check:

Diagonals bisect each other?

AC = BD

Rhombus Check:

Diagonals Perpendicular?

𝐴𝐶̅̅ ̅̅ ⊥ 𝐵𝐷̅̅ ̅̅

Example 5: Identifying a Quadrilateral

The diagonals of ABCD intersect at point N to form four congruent

isosceles triangles: ANB CNB CND AND. What type of

quadrilateral is ABCD? Prove that your answer is correct.

23

QUADRILATERALS

Parallelograms

Quadrilateralssrushingoe.weebly.com/uploads/1/2/5/6/12564201/... · 6-1 Angles of Polygons I CAN… INTERIOR ANGLES OF QUADRILATERALS Classify polygons abased on their sides and angles

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